Spectral Extrema of Bipartite Graphs: Forbidden an Even Cycle of Specified Length

Abstract

A graph is called H-free if the graph has no subgraph being isomorphic to H. The classical Turán problem aims to determine the maximum size of an H-free graph with given order. In 2010, Nikiforov introduced the spectral Turán-type problem: determine the maximum spectral radius of an H-free graph with given order. He also presented the following conjecture: for sufficiently large ν and for k≥2, the graph Kk∨(Kν−k−2¯∪K2) is the unique ν-vertex {C2k+1,C2k+2}-free graph having the largest spectral radius, and Kk∨Kν−k¯ is the unique ν-vertex C2k+2-free graph having the largest spectral radius. Recently, this conjecture was confirmed by Cioabă, Desai, and Tait. In this contribution, we resolve Nikiforov’s even cycle conjecture for bipartite graphs. Let ν and k be two integers satisfying ν≥1 if k=1, and ν≥72224k2k−12(k+1)20(2k−3)2k+1k−1 if k≥2. Then, we show that the complete bipartite graph Kk,ν−k uniquely maximizes the spectral radius over the set of C2k+2-free bipartite graphs of order ν.